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摂動論的量子色力学： 現状と課題

摂動論的量子色力学： 現状と課題. 基研 研究会 (July 31 – Aug. 3, 2006) 「 素粒子物理学の進展 2006 」. J. Kodaira (KEK). Physics at very short distances. Higgs Particle Supersymmetry predicts a host of new massive particles including a dark matter candidate

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摂動論的量子色力学： 現状と課題

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  1. 摂動論的量子色力学： 現状と課題 基研研究会 (July 31 – Aug. 3, 2006) 「素粒子物理学の進展 2006」 J. Kodaira(KEK)

  2. Physics at very short distances • Higgs Particle • Supersymmetry predicts a host ofnew massive particles • including a dark matter candidate • Typical masses ~ 100 GeV/c2 – 1 TeV/c2 • Many other theories of electroweak scale mW,Z = 100 GeV/c2 • make similar predictions: • new dimensions of space-time • new forces • etc. How to reach these particles? →High energy collider

  3. Reality of Nature Hadron – Hadron Collision (LHC) Quark – Gluon Picture From ATLAS Home Page By Huston (CTEQ school 2006)

  4. Indep. → renormalization group eq. Indep. → DGLAP eq. We Want to (Must) Calculate These Events !! What We Can Do? Key Points 1. Asymptotic Freedom in QCD 2. Factorization

  5. Inelastic proton-proton • reactions: 109 / sec • bb pairs 5 106 / sec • tt pairs 8 / sec • W  e n 150 / sec • Z  e e 15 / sec • Higgs (150 GeV) 0.2 / sec • Gluino, Squarks (1 TeV) 0.03 / sec Theoretical Precision Cross Sections and Rates Rates for L = 1034 cm-2 sec-1: Rare Processes: ⇒ high precision experiment ⇒ high precision theoretical prediction From K.Jakobs at CTEQ School ’03

  6. ● Scale （Scheme） Dependence X-Sections do not Depend onμ（R,F,…) BUT Truncation Does Induce Scale Dep. !! ● Edge of Phase Space Multi Scale Problem Large Log Corrections → Resummation ● PDF , PFF Fitting (Non-Pert. Object) Theoretical Ambiguities ● Basic Issue Perturbation ⇔ Asymptotic expansion ⇔ Borel summable ? Renormalon ? Correction expected small ? Data (HERA + TEVATRON + LEP) + Theory used

  7. Try to Reduce Theoretical Ambiguities Error ≦ 10% ¶ Scale (scheme) dependence would be a Good Choice Physically →Look Stability with → LO (Nonsense) → NLO → NNLO → … Two example of higher order calculations ● Higgs production at LHC ● DGLAP kernel at three loop

  8. ● Higgs production at LHC S. Moch and A. Vogt, hep-ph/0508265

  9. ● Higher order cal. of DGLAP Kernel DGLAP Kernel : NNLO = 2-loop Hard Part + 3-loop DGLAP Moch, Vermaseren and Vogt (’04) Theoretical Error: NLO (10-20%) to NNLO (a few – 10%) ►1 – Loop AP Kernel

  10. ► 2 – Loop AP Kernel

  11. ► 3 – Loop AP Kernel

  12. ¶ Edge of Phase Space(Resummation Program) : Finite in Distribution Sense !! →Does Not Mean Good Perturbation In Edge Region of Phase Space Two Scale Problem vs. One Scale Problem Ex. 1 Structure Function Small x ( s >> Q2) , Large x ( s << Q2) Ex. 2 Drell – Yan Type Process Threshold Logs Recoil or QTLogs Large αS ln 2 ( Ratio of Two Scales) Appear

  13. Structure of PT with Two Different Scales Putting → truncate at a fixed Order → resum Terms Rearrange Perturbative Series such that Leading Log (LL) Next to Leading Log (NLL) (NNLL)

  14. QT Resummation • Collins and Soper (’81) Kodaira and Trentadue (’82) Higgs QT Distribution with Error Band (NNLO vs. NNLL) Comparison of NLL+LO and NNLL+NLO Bands Bozzi, Catani , de Florian and Grazzini (’05)

  15. Higher Order Calculations at (N)NLO 2 → 2(3) Amplitude : Loop calculation Master Integral: Almost Finished(?) Good shape for simple (signal) process Multi–Leg (2 → n ) Amplitude (multi particle fs.) ex. Gluino difficult even at tree level !! Cascade Decay to Light (SM) Particles Backgrounds (and many signals) require detailed understanding of scattering amplitudes for many ultra-relativistic particles

  16. A realistic NLO wishlist for multi-particle final states Les Houches 2005

  17. Wishlist from Tevatron (Run II)

  18. Feynman Told Us How to Calculate - in Principle Procedure ( Feynman Rule ): 1. Draw Diagrams Too Many Diagrams 2. Calculate M Too Many Terms 3. Calculate |M|2 Too Many Terms and Kinematic Variables 4. Take Sum and/or Average ∑ |M|2 ↓ Answer: in terms of Lorentz Product of 4 – Momentum (Many terms Cancel Each Others at 3.) Feynman rules, while very general, but not optimized for multi-leg processe Consider Tree Level Multi–Leg Amplitude Example: # of Feynman Diagrams for gg → ng at tree !! n = 2 3 4 5 6 7 8 # of Diagrams 4 25 220 2485 34300 559405 10525900

  19. Feynman diagrams for QCD are “too complicated” Too Many Terms from only 10 diagrams!

  20. New Technique for Calculating Amplitudes Brief history of development in calculational technique ● Spinor Helicity Method MHV amplitude (Parke & Taylor ’86) Off-shell recursion relation (Berends & Giele ’88) 1-loop extension make use of Supersymmetry and/or unitarity (Bern, Dixon & Kosower ’93) ● String in Twistor Space Twistor string ⇔ QCD (Witten ’04) MHV rule (Cachazo, Svrček & Witten ’04) ● On-Shell Recursion Relation On-shell recursion relation – string based – (Britto, Cachazo & Feng ’05) On-shell recursion relation – field theory based – (Britto, Cachazo, Feng & Witten ’05)

  21. ¶ Spinor Helicity Method Calculate M in terms of Spinor Product For Massless Spinor (Fermion) Can DefineChiral or Helicity Eigenstate Use Shorthand Notation Spinor Product In Weyl (2-dimensional) representation

  22. Important Identity for Spinor Product This Means Spinor Product is Square Root of Lorentz Product up to Phase Excellent variables for helicity amplitudes scalars gauge theory Angular Momentum Cons.

  23. Polarization Vector for Massless Vector Boson (gluon) Can also be Expressed by Spinor Product!! In Weyl representation q is Arbitrary Massless Vector Called Reference Momentum Satisfies All Properties of Polarization Vector Additional useful identities Change of q induces term proportional to so ｆor Gauge Invariant Amplitude We can choose any q !! →Judicious Choice of q to Simplify Calculation

  24. ● Amplitude in Spinor Helicity Method For Simplicity, Consider Pure Yang-Mills (Gluon) Sector Strategy to Simplify Calculation 1. Decompose Amplitude into the Sum of Independent (in Color Space) Sub-amplitude 2. Sub-amplitude is Gauge Invariant →Judicious Choice of q to Simplify Calculation n-Gluon Scattering amplitude with Color External Momenta Helicities One Can Show where g : Coupling Const, (1,2,…,n)’ : Non-cyclic Permutation

  25. Color Factors are Independent • →Sub-amplitudes are Gauge Invariant • →Judicious Choice of q to Simplify Calculation • (2) “Dual” Ward Identity Proof: (rough Idea) Formula for is Valid for any Nc put Nc = 1, i.e. Ta = 1 ⇔U(1) theory ↓ Gluon does not have Self-coupling → cf. These Properties Can be Easily Proved using String Theory (taking zero-slope limit)

  26. MHV (Maximally Helicity Violating) Amplitude by Convention, all Particles are Out-going MHV Amplitude the Name of “MHV” in Real Process, some (2) of Particles Should be in the Initial State　（In-Coming) So, the above means Since out-going　 　 ←　in-coming but

  27. Next Possibility one of particle, say “1” has opposite Helicity but proof of above An=0 (1) Explicit Calculation (See Next) (2) use of Super Symmetric Theory

  28. Consider There are at Most n-2 Vertices at Tree Level Three Gluon Vertex: one Momentum Vector Four Gluon Vertex: no Momentum Vector On the other Hand, there are n Polarization Vectors Since An Does not Have Lorentz Suffix, there must be at least one Contraction by Choosing the Same Reference Momentum for All for choose

  29. Parke – Taylor Formula (1986) so, non-vanishing MHV amplitude is two of Particleshave Opposite Helicities MHV Amplitude

  30. ^ n n ● ● ● 1 ^ 1 3 ● 2 ● ● h k+1 = -h k ● ● ● 3 2 ¶ On-Shell Recursion Relation Big Problems of How to Calculate non-MHV Amplitude!! Great Progress in last few Years!! Britto, Cachazo, Feng, Nucl.Phys.B715(2005)499 Ak+1 and An-k+1 are on-shell tree amplitudes with fewer legs, evaluated with momenta shifted by a complex amount Trees are recycled into trees!

  31. Proof of on-shell recursion relations Britto, Cachazo, Feng, Witten, Phys.Rev.Lett.94(2005)181602 Simple, general: Residue (Cauchy) theorem Consider a tree n gluon amplitude Pick up 2 Arbitrary gluons : say, 1 and n Inject complex momentum at leg 1, remove it at leg n. with arbitrary Complex z therefore becomes an Analytic Function of z and is Physical Amplitude

  32. e.g. one Diagram for 5-point Amplitude 3 3 4 4 → ● 2 ● ● 2 ● ● ● ^ 5 5 ^ 1 1 Singularity ⇔ Singularity of (at tree level) Comes from the Propagator of Gluon in the Amplitude with Shifted Momentum Momentum of propagator at tree level is always a sum of (adjacent) external momenta where

  33. ^ n ^ 1 In General k k+1 ・ ・ ・ ・ ・ ・ -h h 3 2 and Singularity w.r.t. z is at where

  34. This Means is Rational Function with Single Pole at Cauchy’s Theorem: if (one can check at tree) then finally Key Points: namely “sub-amplitudes” are “on-shell”

  35. Example for 6-gluon Amplitude 220 Feynman diagrams for gggggg Helicity + color + MHV results + symmetries 3 recursive diagrams Each Blocks are MHV Amplitudes which are Known Note: 3-Point Function is Special!!

  36. Simpler than form found in 1980s Mangano, Parke, Xu (1988) Simple final form

  37. Relative Simplicity Grows with n Berends, Giele, Kuijf (1990)

  38. = Bern, Del Duca, Dixon , Kosower (2004)

  39. Present status of this field (formulation) (B-Theory) So many papers now from not so many authors ・ inclusion of fermion (quark) ・ extension to (1-) loop level (currently big activities) Bern, Dixon & Kosower et al. ・ inclusion of massive particle not practical interest (?) for pure EW (many CP programs e.g. GRACE at KEK) ・ automatization of program in CP (to Xs section) how to check the correctness of (numerical) results gauge (parameter) independence is a strong check but B-theory makes use of it from the beginning ・ further understanding of gauge theory (QCD) ・ ………

  40. Conclusion QCD is a Theory of Strong Interaction Non-abelin Gauge Theory One of the “most” beautiful The. in 20th Century Forced to solve (at least) two problems now 1. do calculation for practical purpose (LHC, ILC, …) 2. reveal dynamics of gauge theory using the simplest but the most difficult one namely QCD

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